Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2190 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#2190 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #447 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
590 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 590 conjugacy classes of subgroups — too many to draw. Only its 127 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2);(4,8)(5,11)(6,14)(9,16)(10,17)(12,15)(13,18);(3,4)(5,12)(6,15)(7,9)(8,11)(10,14)(13,17)(16,18);(5,13)(8,16)(11,18)(12,17)40 conjugacy classes — view the full character table.
| maximal subgroup of | #137392, #137395, #137397, … |
| maximal normal subgroup of | #137392, #137395, #137397, … |
| maximal quotient of | C2×#12102, C2×#12103, C2×#12104, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26516);